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Additional resources for An Operator Semigroup in Mathematical Genetics

Example text

The archetype of complete normed spaces are real numbers: all Cauchy sequences in R converge. In contrast, the set of rational numbers Q does not posses this property. For example, we may find a sequence of rational numbers converging to π : this sequence is obviously a Cauchy sequence, and on the other hand cannot converge to any rational number, since it does converge to π ∈ Q. ), consider the space C[0, 2] of continuous functions on the interval [0, 2]. It is easy to see that, when equipped with the norm, 2 x = |x(s)| ds, 0 C[0, 2] is a linear normed space (the sum of x, y ∈ C[0, 2] is defined by (x + y)(s) = x(s) + y(s) and the product of α ∈ R and x ∈ C[0, 2] is defined by (αx)(s) = αx(s), s ∈ [0, 2]).

Finally, for t < t0 we have P(t)x − P(t0 )x = P(t)x − P(t)P(t0 − t)x = P(t)[x − P(t0 − t)x] ≤ x − P(t − t0 )x , since the norm of P(t) is 1. 25). 25) implies in particular that functions [0, ∞) t → pi, j (t) are continuous: to see this it is enough to take x = ei and recall that strong convergence in l 1 forces convergence of coordinates. 25) make {P(t), t ≥ 0} a strongly continuous semigroup. 26) where a, b ≥ 0, a + b > 0 are real numbers. Here, the semigroup property can be checked by hand, and the strong continuity follows by continuity of coordinates since the index set I is finite (Comp.

Moreover, it suffices to show that 1− pti,i (t) vi (t) converges to q1i vi . This, however, follows by Schéffe’s theorem since all the vectors involved here are distributions, and they converge coordinate-wise. To recapitulate: if the state-space is infinite, instead of a Kolmogorov matrix, we may consider a closed linear operator Q that to some extend mimics this matrix. This closed operator is termed the generator of the semigroup (of transition matrices). The name comes from the fact that, as the general theory of semigroups of operators shows, this single operator contains all the information on the semigroup, or, put otherwise, there is a way of obtaining all P(t)’s from Q.